By well-known results in graph signal processing [1]}, simultaneous diagonalizability of the system matrices \((A,B)\)  and the GMD \(S\)  impliesUnder the assumption that \(S\)  corresponds to a finite graph and has all distinct eigenvalues. On a high level, the result follows directly from the Cayley-Hamilton theorem. that they can be written as matrix polynomials of \(S\)  of degrees at most \(N-1\) ,
\( A = \sum _{k=0}^{N-1}h_{A,k}S^k \quad , \quad B = \sum _{k=0}^{N-1}h_{B,k}S^k.\) 
